Standard notation or name for the measure defined by an integral over a function

Question

Let $(\Omega,\mathcal{F},\mu)$ be a measure space. For a measurable function $f \geq 0$ one can define a measure $\nu$ by $$ \nu(M) : = \int_M f d \mu = \int_{\Omega} f \mathbb{1}_{M} d \mu $$ for $M \in \mathcal{F}$. Is there a standard notation/name for the measure $\nu$? I think I saw something like $f \odot \mu$ but I am not sure.

Answer 1

The name given to a map from a function to some field (usually the reals) obtained by using that function in some sort of operator is a functional (or sometimes a functor).

The notation often uses square brackets, as in $\nu_M[f]$. The branch of math this often appears in is the calculus of variations.

Answer 2

I have seen it used in the form $f\odot \mu$ in probability theory, of course for the measure that is absolutely continuous with respect to $\mu$ with Radon-Nikodym derivative $f$. But it is not a very nice notation, since it misses the right form $d\mu$.

It is common (without using your notation!) in calculus of variations (Young measures), ergodic theory (equilibrium measures), multifractal analysis (Gibbs measures), delay equations (bounded variation functions and Riemann-Stieltjes), etc. But I recall no notation being used that could be widely accepted.